With the release of a new Star Wars movie, there is renewed interest in the original trilogy. Unfortunately for the new viewer, these films come in several flavors, and choosing which ones to watch can be confusing. I'm going to focus on four different groupings before telling what my recommended versions are:
1) The original films, as released from 1977-83
2) The 1997 special editions
3) The 2011 Blu-ray releases
4) Fan Edits
Fan edits, if you are unaware, are attempts by fans to put their own stamp on a film's story. Technology has advanced to the point that anyone with a computer, some software, and a modicum of talent can craft a personal version of a movie.
I'm going to post in reverse order, as I suspect my choices get more controversial that way.
Return of the Jedi
First of all, the one to not watch is the Blu-ray version. The inclusion of Hayden Christensen's Anakin force ghost makes no sense whatsoever. It's not like Alec Guiness turned into Ewan MacGregor. And how would Luke know who that kid is anyway? He's only ever seen old Anakin, played nicely by Sebastian Shaw.
George Lucas always claimed he didn't have the money or technology to make Episode IV the way he wanted, but by the time 1983 rolled around he had both: all the money he needed, and a state of the art special effects studio. Which version to watch boils down to one question: Do you like Yub Nub?
If you enjoy the Ewok celebration song, watch the original 1983 version. It looks as good today as it did in 1983. Seek out Harmy's Despecialized Version for a superior presentation of the 1983 film.
If you hate Yub Nub, watch the 1997 special edition (if you can find it). It doesn't really change that much from the 1983 version (a different Jabba palace song, a beak on the sarlacc, and a few other things -- nothing major).
The Empire Strikes Back
Unlike episodes IV and VI, Empire was lightly touched by the special edition people. Plus, the changes to the film are quite nice. The extra cloud city scenes and windows really serve to open up that part of the film, while the original feels too white and claustrophobic.
My recommendation is to watch the Blu-ray. This version has the additional advantage of using Ian McDiarmid for the emperor, maintaining continuity with Jedi and the prequels.
(aka Star Wars Episode IV: A New Hope)
Many people will say to watch the original version, but Lucas is right about one thing: he didn't have the technology to realize his vision. The effects while revolutionary in 1977 look primitive today, and worse, do not align with the rest of the saga. In the worst case, cardboard cutouts are used for people in the celebration scene.
However, in the process of fixing these things, Lucas messed up a lot as well. Most infamously is the "Han Shot First" bit, but also annoying are slapstick comedic elements added throughout, and an added Jabba scene that fails on every level: it looks bad, even in the 2011 version; it simply restates the earlier Greedo scene; and it destroys the Jabba reveal in Episode VI.
That leaves us in the hands of the Fan Editors, and there is one fan editor that has gone above and beyond to make a seriously special edition of the film: Adywan. Seek out Adywan's "Star Wars Revisited" fan edit. He fixes all the good stuff, takes out the bad stuff, and makes Episode IV fit with the rest of the saga.